Algebras, Automata and Logic for Languages of Labeled Birooted Trees

  • David Janin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)


In this paper, we study the languages of labeled finite birooted trees: Munn’s birooted trees extended with vertex labeling. We define a notion of finite state birooted tree automata that is shown to capture the class of languages that are upward closed w.r.t. the natural order and definable in Monadic Second Order Logic. Then, relying on the inverse monoid structure of labeled birooted trees, we derive a notion of recognizable languages by means of (adequate) premorphisms into finite (adequately) ordered monoids. This notion is shown to capture finite boolean combinations of languages as above. We also provide a simple encoding of finite (mono-rooted) labeled trees in an antichain of labeled birooted trees that shows that classical regular languages of finite (mono-rooted) trees are also recognized by such premorphisms and finite ordered monoids.


Inverse Semigroup Order Logic Natural Order Regular Language Tree Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blumensath, A.: Recognisability for algebras of infinite trees. Theor. Comput. Sci. 412(29), 3463–3486 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bojanczyk, M., Straubing, H., Walukiewicz, I.: Wreath products of forest algebras, with applications to tree logics. Logical Methods in Computer Science 8(3) (2012)Google Scholar
  3. 3.
    Bojańczyk, M., Walukiewicz, I.: Forest algebras. In: Logic and Automata, pp. 107–132 (2008)Google Scholar
  4. 4.
    Cornock, C., Gould, V.: Proper two-sided restriction semigroups and partial actions. Journal of Pure and Applied Algebra 216, 935–949 (2012)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ésik, Z., Weil, P.: On logically defined recognizable tree languages. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 195–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Janin, D.: Quasi-recognizable vs MSO definable languages of one-dimensional overlapping tiles. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 516–528. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Janin, D.: Walking automata in the free inverse monoid. Technical Report RR-1464-12 (revised April 2013), LaBRI, Université de Bordeaux (2012)Google Scholar
  8. 8.
    Janin, D.: On languages of one-dimensional overlapping tiles. In: van Emde Boas, P., Groen, F.C.A., Italiano, G.F., Nawrocki, J., Sack, H. (eds.) SOFSEM 2013. LNCS, vol. 7741, pp. 244–256. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Janin, D.: Overlapping tile automata. In: Bulatov, A. (ed.) CSR 2013. LNCS, vol. 7913, pp. 431–443. Springer, Heidelberg (2013)Google Scholar
  10. 10.
    Kellendonk, J., Lawson, M.V.: Tiling semigroups. Journal of Algebra 224(1), 140–150 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lawson, M.V.: McAlister semigroups. Journal of Algebra 202(1), 276–294 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Perrin, D., Pin, J.-E.: Semigroups and automata on infinite words. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups. NATO Advanced Study Institute, pp. 49–72. Kluwer Academic (1995)Google Scholar
  13. 13.
    Pin, J.-E.: Relational morphisms, transductions and operations on languages. In: Pin, J.E. (ed.) LITP 1988. LNCS, vol. 386, pp. 34–55. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  14. 14.
    Pin, J.-E.: Finite semigroups and recognizable languages: an introduction. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups. NATO Advanced Study Institute, pp. 1–32. Kluwer Academic (1995)Google Scholar
  15. 15.
    Pin, J.-.E.: Syntactic semigroups. In: Handbook of Formal Languages, ch. 10, vol. I, pp. 679–746. Springer (1997)Google Scholar
  16. 16.
    Scheiblich, H.E.: Free inverse semigroups. Semigroup Forum 4, 351–359 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Shelah, S.: The monadic theory of order. Annals of Mathematics 102, 379–419 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Thomas, W.: Languages, automata, and logic. In: Handbook of Formal Languages, ch. 7, vol. III, pp. 389–455. Springer (1997)Google Scholar
  19. 19.
    Wilke, T.: An algebraic theory for regular languages of finite and infinite words. Int. J. Alg. Comput. 3, 447–489 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • David Janin
    • 1
  1. 1.LaBRI UMR 5800Université de BordeauxTalenceFrance

Personalised recommendations